UK Property Law
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The document you are referring to is called a declaration of trust and would usually be drafted then executed upon completion by the conveyancer acting in the purchase.
The terms of the declaration of trust - whether they are to receive a fixed sum or a proportional one depending on the increase in equity - would be for you to agree between each other and then pass appropriate instructions to the conveyancer on how it should be drafted.
In the absence of any such declaration of trust it is presumed that each party holds a 50% share each in the equity of the property.
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